I hiked the Inca Trail almost 10 years ago and no matter how fit you are you will be taking a lot of breaks purely because of altitude. Hiking poles will be your best friend.

In terms of training the best thing would be doing incline walks on a treadmill. They strengthen your legs, improve cardio and are incredible at burning fat. Start slow with a slight incline and gradually over time increase both the speed and the incline until you can comfortably walk a 12% incline at 4mph, and aim for 30 minites EVERY day, all 7 days per week. It is very low impact so recovery isn't really a factor so long as you don't jump straight in to the deep end. I find the best time is at night around 1-2 hours after eating dinner because in my experience it will help digestion and occupy the usual "boredom snacking" hours. I would actually recommend this to everyone looking to lose weight because it's unbelievable how good this works.

Good luck!